NONEXISTENCE AND SHORT TIME ASYMPTOTIC BEHAVIOR OF SOURCE-TYPE SOLUTION FOR POROUS MEDIUM EQUATION WITH CONVECTION IN ONE-DIMENSION

摘要

In this paper we consider the following equation u_t= (u~m)_(xx) + (u~n)_x, (x, t) ∈ R x (0, ∞) with a Dirac measure as initial data, i.e., u(x,0) = δ(x). The solution of the Cauchy problem is well-known as source-type solution. In the recent work [11] the author studied the existence and uniqueness of such kind of singular solutions and proved that there exists a number n_0 = m + 2 such that there is a unique source-type solution to the equation when 0 ≤ n < n_0. Here our attention is focused on the nonexistence and asymptotic behavior near the origin for a short time. We prove that n_0 is also a critical number such that there exits no source-type solution when n ≥ n_0 and describe the short time asymptotic behavior of the source-type solution to the equation when 0 ≤ n < n_0. Our result shows that in the case of existence and for a short time, the source-type solution of such equation behaves like the fundamental solution of the standard porous medium equation when 0 ≤ n < m + 1, the unique self-similar source-type solution exists when n = m + 1, and the solution does like the nonnegative fundamental entropy solution in the conservation law when m + 1 < n < n_0, while in the case of nonexistence the singularity gradually disappears when n ≥ n_0 that the mass cannot concentrate for a short time and no such a singular solutions exists. The results of previous work [11] and this paper give a perfect answer to such topical researches.